1 | ////////////////////////////////////////////////////////////////////////////// |
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2 | version="$Id: jacobson.lib,v 1.4 2008/12/01 20:51:16 levandov Exp $"; |
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3 | category="System and Control Theory"; |
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4 | info=" |
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5 | LIBRARY: jacobson.lib Algorithms for Smith and Jacobson Normal Form |
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6 | AUTHOR: Kristina Schindelar, Kristina.Schindelar@math.rwth-aachen.de |
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7 | |
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8 | THEORY: We work over a principal ideal domain R. We suppose R to be either |
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9 | @* the commutative polynomial ring in one variable or the Ore localization of |
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10 | @* the first Weyl algebra with respect to S = K[x]\{0} (i.e. K(x)<D|D*x=x*D+1>) |
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11 | @* Under these assumptions on R, given a rectangular matrix M over R, one can |
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12 | @* compute unimodular matrices U, V such that U*M*V=D is a diagonal matrix. |
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13 | @* Depending on the ring, the diagonal entries of D have certain properties. |
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14 | @* Over a commutative ring in one variable, the matrix D is unique and called |
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15 | @* the Smith Normal Form of M, whereas in the non-commutative case D is not |
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16 | @* unique and a weak Jacobson Normal Form of M is computed. |
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17 | |
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18 | TODO: WHAT ABOUT UNIQUENESS OF JACOBSON NORMAL FORM? WHICH ONE IS RETURNED? |
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19 | |
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20 | PROCEDURES: |
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21 | smith(M[,eng1,eng2]); computes the Smith Normal Form |
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22 | jacobson(M[,eng]); computes a weak Jacobson Normal Form |
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23 | |
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24 | SEE ALSO: control_lib |
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25 | "; |
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26 | |
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27 | LIB "poly.lib"; // TODO: WHAT FOR? |
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28 | LIB "involut.lib"; // TODO: WHAT FOR? |
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29 | |
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30 | /////////////////////////////////////////////////////////////////////////////// |
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31 | |
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32 | // TODO: IT IS BETTER TO WORK WITH MODULES INSTEAD OF MATRICES!!! |
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33 | // PLEASE SWITCH MATRICES TO MODULES! |
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34 | |
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35 | // TODO: IS THE ALGORITHM WELL KNOWN OR IS IT DUE TO SOME BOOK??? |
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36 | // ADD REFERENCE IS SO. |
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37 | proc smith(matrix MA, list #) |
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38 | "USAGE: smith(M[, eng1, eng2]); M matrix, eng1 and eng2 are optional integers |
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39 | RETURN: matrix or list, either a matrix D or a list containing matrices {U,D,V}, |
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40 | such that U*M*V = D, where D is diagonal (the Smith Normal Form of M) |
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41 | and the matrices U and V are unimodular. |
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42 | ASSUME: The current ring is assumed to be the commutative polynomial ring in |
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43 | one variable |
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44 | NOTE: If the optional integer eng1 is non-zero, returns the list {U,D,V}, |
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45 | @* otherwise only the matrix, the Smith Normal Form of M is returned. |
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46 | @* The optional integer eng2 determines the engine, that computes the GB: |
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47 | @* 0 means 'std' (default), 1 means 'groebner' and 2 means 'slimgb'. |
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48 | @* If @code{printlevel}=1, progress debug messages will be printed, |
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49 | @* if @code{printlevel}>=2, all the debug messages will be printed. |
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50 | EXAMPLE: example smith; shows examples |
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51 | " |
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52 | { |
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53 | def R = basering; |
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54 | |
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55 | // TODO: ADD ASSUMPTION CHECKS!!! |
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56 | |
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57 | int eng = 0; |
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58 | int BASIS = 0; |
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59 | |
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60 | if ( size(#) > 0 ) |
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61 | { |
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62 | if (typeof(#[1])=="int") |
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63 | { |
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64 | eng=#[1]; // zero can also happen |
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65 | } |
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66 | |
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67 | if (size(#) > 1) |
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68 | { |
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69 | if (typeof(#[2])=="int") |
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70 | { |
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71 | BASIS=#[2]; |
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72 | } |
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73 | } |
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74 | } |
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75 | |
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76 | |
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77 | int ROW=ncols(MA); |
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78 | int COL=nrows(MA); |
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79 | |
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80 | //generate a module consisting of the columns of MA |
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81 | module m = MA; |
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82 | int i; |
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83 | |
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84 | //if MA eqauls the zero matrix give back MA |
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85 | if ( (size(m)==0) and (size(transpose(m))==0) ) |
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86 | { |
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87 | return(list( matrix(freemodule(COL)), MA, matrix(freemodule(ROW)) )); |
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88 | } |
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89 | |
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90 | if(eng > 0) |
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91 | { |
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92 | list rueckLI=diagonal_with_trafo(R,MA,BASIS); |
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93 | list rueckLII=divisibility(rueckLI[2]); |
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94 | matrix CON=divideByContent(rueckLII[2]); |
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95 | list rueckL=CON*rueckLII[1]*rueckLI[1], CON*rueckLII[2], rueckLI[3]*rueckLII[3]; |
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96 | return(rueckL); |
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97 | } |
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98 | else |
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99 | { |
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100 | matrix rueckm=diagonal_without_trafo(R,MA,BASIS); |
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101 | list rueckL=divisibility(rueckm); |
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102 | matrix CON=divideByContent(rueckm); |
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103 | rueckm=CON*rueckL[2]; |
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104 | return(rueckm); |
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105 | } |
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106 | } |
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107 | example |
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108 | { "EXAMPLE:"; echo = 2; |
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109 | ring r = 0,x,Dp; |
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110 | matrix m[3][2]=x, x^4+x^2+21, x^4+x^2+x, x^3+x, 4*x^2+x, x; |
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111 | print(m); // M |
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112 | print(smith(m)); // Smith Normal Form of M |
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113 | list S=smith(m,1); |
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114 | // TODO: BUG THE FOLLOWING GIVES A DIFFERENT RESULT TO ABOVE! |
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115 | print(S[2]); // Smith Normal Form D of M |
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116 | // TODO: ARE U AND V REALLY UNIMODULAR? |
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117 | S[1]; // U |
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118 | det(S[1]); |
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119 | S[3]; // V |
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120 | det(S[3]); |
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121 | print(S[1]*m*S[3] - S[2]); // check that U*M*V = D |
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122 | print(smith(matrix(0))); // Smith Normal Form of 0 |
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123 | |
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124 | } |
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125 | |
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126 | // TODO: THIS PROCEDURE SEEMS TOO COMPLICATED, THUS PLEASE |
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127 | // TODO: 1. ADD HELP WITH ASSUMPTIONS AND RETURN DESCRIPTION |
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128 | // TODO: 2. ADD STEP-BY-STEP COMMENTS INTO THE CODE |
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129 | |
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130 | static proc diagonal_with_trafo( R, matrix MA, int B) |
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131 | { |
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132 | |
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133 | int ppl = printlevel-voice+2; |
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134 | |
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135 | int BASIS=B; |
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136 | int ROW=ncols(MA); |
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137 | int COL=nrows(MA); |
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138 | module m=MA[1]; |
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139 | int i,j,k; |
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140 | for(i=2;i<=ROW;i++) |
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141 | { |
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142 | m=m,MA[i]; |
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143 | } |
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144 | |
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145 | |
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146 | //add zero rows or columns |
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147 | //add zero rows or columns |
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148 | int adrow=0; |
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149 | for(i=1;i<=COL;i++) |
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150 | { |
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151 | k=0; |
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152 | for(j=1;j<=ROW;j++) |
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153 | { |
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154 | if(MA[i,j]!=0){k=1;} |
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155 | } |
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156 | if(k==0){adrow++;} |
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157 | } |
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158 | |
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159 | m=transpose(m); |
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160 | for(i=1;i<=adrow;i++){m=m,0;} |
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161 | m=transpose(m); |
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162 | |
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163 | list RINGLIST=ringlist(R); |
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164 | list o="C",0; |
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165 | list oo="lp",1; |
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166 | list ORD=o,oo; |
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167 | RINGLIST[3]=ORD; |
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168 | def r=ring(RINGLIST); |
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169 | setring r; |
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170 | //fix the required ordering |
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171 | map MAP=R,var(1); |
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172 | module m=MAP(m); |
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173 | |
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174 | int flag=1; ///////////////counts if the underlying ring is r (flag mod 2 == 1) or ro (flag mod 2 == 0) |
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175 | |
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176 | module TrafoL=freemodule(COL); |
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177 | module TrafoR=freemodule(ROW); |
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178 | module EXL=freemodule(COL); //because we start with transpose(m) |
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179 | module EXR=freemodule(ROW); |
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180 | |
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181 | option(redSB); |
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182 | option(redTail); |
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183 | module STD_EX; |
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184 | module Trafo; |
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185 | |
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186 | int s,st,p,ff; |
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187 | |
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188 | module LT,TSTD; |
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189 | module STD=transpose(m); |
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190 | int finish=0; |
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191 | int fehlpos; |
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192 | list pos; |
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193 | matrix END; |
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194 | matrix ENDSTD; |
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195 | matrix STDFIN; |
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196 | STDFIN=STD; |
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197 | list COMPARE=STDFIN; |
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198 | |
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199 | while(finish==0) |
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200 | { |
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201 | dbprint(ppl,"Going into the while cycle"); |
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202 | |
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203 | if(flag mod 2==1) |
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204 | { |
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205 | STD_EX=EXL,transpose(STD); |
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206 | } |
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207 | else |
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208 | { |
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209 | STD_EX=EXR,transpose(STD); |
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210 | } |
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211 | dbprint(ppl,"Computing Groebner basis: start"); |
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212 | dbprint(ppl-1,STD); |
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213 | STD=engine(STD,BASIS); |
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214 | dbprint(ppl,"Computing Groebner basis: finished"); |
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215 | dbprint(ppl-1,STD); |
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216 | |
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217 | if(flag mod 2==1){s=ROW; st=COL;}else{s=COL; st=ROW;} |
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218 | dbprint(ppl,"Computing Groebner basis for transformation matrix: start"); |
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219 | dbprint(ppl-1,STD_EX); |
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220 | STD_EX=transpose(STD_EX); |
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221 | STD_EX=engine(STD_EX,BASIS); |
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222 | dbprint(ppl,"Computing Groebner basis for transformation matrix: finished"); |
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223 | dbprint(ppl-1,STD_EX); |
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224 | |
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225 | //////// split STD_EX in STD and the transformation matrix |
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226 | STD_EX=transpose(STD_EX); |
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227 | STD=STD_EX[st+1]; |
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228 | LT=STD_EX[1]; |
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229 | |
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230 | ENDSTD=STD_EX; |
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231 | for(i=2; i<=ncols(ENDSTD); i++) |
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232 | { |
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233 | if (i<=st) |
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234 | { |
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235 | LT=LT,STD_EX[i]; |
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236 | } |
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237 | if (i>st+1) |
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238 | { |
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239 | STD=STD,STD_EX[i]; |
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240 | } |
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241 | } |
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242 | |
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243 | STD=transpose(STD); |
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244 | LT=transpose(LT); |
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245 | |
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246 | ////////////////////// compute the transformation matrices |
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247 | |
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248 | if (flag mod 2 ==1) |
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249 | { |
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250 | TrafoL=transpose(LT)*TrafoL; |
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251 | } |
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252 | else |
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253 | { |
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254 | TrafoR=TrafoR*LT; |
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255 | } |
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256 | |
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257 | |
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258 | STDFIN=STD; |
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259 | /////////////////////////////////// check if the D-th row is finished /////////////////////////////////// |
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260 | COMPARE=insert(COMPARE,STDFIN); |
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261 | if(size(COMPARE)>=3) |
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262 | { |
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263 | if(STD==COMPARE[3]){finish=1;} |
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264 | } |
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265 | ////////////////////////////////// change to the opposite module |
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266 | TSTD=transpose(STD); |
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267 | STD=TSTD; |
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268 | flag++; |
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269 | dbprint(ppl,"Finished one while cycle"); |
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270 | } |
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271 | |
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272 | |
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273 | if (flag mod 2!=0) { STD=transpose(STD); } |
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274 | |
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275 | |
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276 | //zero colums to the end |
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277 | matrix STDMM=STD; |
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278 | pos=list(); |
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279 | fehlpos=0; |
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280 | while( size(STD)+fehlpos-ncols(STDMM) < 0) |
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281 | { |
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282 | for(i=1; i<=ncols(STDMM); i++) |
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283 | { |
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284 | ff=0; |
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285 | for(j=1; j<=nrows(STDMM); j++) |
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286 | { |
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287 | if (STD[j,i]==0) { ff++; } |
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288 | } |
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289 | if(ff==nrows(STDMM)) |
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290 | { |
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291 | pos=insert(pos,i); fehlpos++; |
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292 | } |
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293 | } |
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294 | } |
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295 | int fehlposc=fehlpos; |
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296 | module SORT; |
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297 | for(i=1; i<=fehlpos; i++) |
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298 | { |
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299 | SORT=gen(2); |
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300 | for (j=3;j<=ROW;j++) |
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301 | { |
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302 | SORT=SORT,gen(j); |
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303 | } |
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304 | SORT=SORT,gen(1); |
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305 | STD=STD*SORT; |
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306 | TrafoR=TrafoR*SORT; |
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307 | } |
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308 | |
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309 | //zero rows to the end |
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310 | STDMM=transpose(STD); |
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311 | pos=list(); |
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312 | fehlpos=0; |
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313 | while( size(transpose(STD))+fehlpos-ncols(STDMM) < 0) |
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314 | { |
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315 | for(i=1; i<=ncols(STDMM); i++) |
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316 | { |
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317 | ff=0; for(j=1; j<=nrows(STDMM); j++) |
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318 | { |
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319 | if(transpose(STD)[j,i]==0){ ff++;}} |
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320 | if(ff==nrows(STDMM)) { pos=insert(pos,i); fehlpos++; } |
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321 | } |
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322 | } |
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323 | int fehlposr=fehlpos; |
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324 | |
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325 | for(i=1; i<=fehlpos; i++) |
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326 | { |
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327 | SORT=gen(2); |
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328 | for(j=3;j<=COL;j++){SORT=SORT,gen(j);} |
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329 | SORT=SORT,gen(1); |
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330 | SORT=transpose(SORT); |
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331 | STD=SORT*STD; |
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332 | TrafoL=SORT*TrafoL; |
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333 | } |
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334 | |
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335 | setring R; |
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336 | map MAPinv=r,var(1); |
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337 | module STD=MAPinv(STD); |
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338 | module TrafoL=MAPinv(TrafoL); |
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339 | matrix TrafoLM=TrafoL; |
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340 | module TrafoR=MAPinv(TrafoR); |
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341 | matrix TrafoRM=TrafoR; |
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342 | matrix STDM=STD; |
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343 | |
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344 | //Test |
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345 | if(TrafoLM*m*TrafoRM!=STDM){ return(0); } |
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346 | |
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347 | list RUECK=TrafoRM; |
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348 | RUECK=insert(RUECK,STDM); |
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349 | RUECK=insert(RUECK,TrafoLM); |
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350 | return(RUECK); |
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351 | } |
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352 | |
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353 | |
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354 | |
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355 | // TODO: THIS PROCEDURE SEEMS TOO COMPLICATED, THUS PLEASE |
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356 | // TODO: 1. ADD HELP WITH ASSUMPTIONS AND RETURN DESCRIPTION |
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357 | // TODO: 2. ADD STEP-BY-STEP COMMENTS INTO THE CODE |
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358 | |
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359 | static proc divisibility(matrix M) |
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360 | { |
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361 | matrix STDM=M; |
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362 | int i,j; |
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363 | int ROW=nrows(M); |
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364 | int COL=ncols(M); |
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365 | module TrafoR=freemodule(COL); |
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366 | module TrafoL=freemodule(ROW); |
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367 | module SORT; |
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368 | matrix TrafoRM=TrafoR; |
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369 | matrix TrafoLM=TrafoL; |
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370 | list posdeg0; |
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371 | int posdeg=0; |
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372 | int act; |
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373 | int Sort=ROW; |
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374 | if(size(std(STDM))!=0) |
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375 | { while( size(transpose(STDM)[Sort])==0 ){Sort--;}} |
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376 | |
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377 | for(i=1;i<=Sort ;i++) |
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378 | { |
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379 | if(leadexp(STDM[i,i])==0){posdeg0=insert(posdeg0,i);posdeg++;} |
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380 | } |
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381 | //entries of degree 0 at the beginning |
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382 | for(i=1; i<=posdeg; i++) |
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383 | { |
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384 | act=posdeg0[i]; |
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385 | SORT=gen(act); |
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386 | for(j=1; j<=COL; j++){if(j!=act){SORT=SORT,gen(j);}} |
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387 | STDM=STDM*SORT; |
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388 | TrafoRM=TrafoRM*SORT; |
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389 | SORT=gen(act); |
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390 | for(j=1; j<=ROW; j++){if(j!=act){SORT=SORT,gen(j);}} |
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391 | STDM=transpose(SORT)*STDM; |
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392 | TrafoLM=transpose(SORT)*TrafoLM; |
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393 | } |
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394 | |
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395 | poly G; |
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396 | module UNITL=freemodule(ROW); |
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397 | matrix GCDL=UNITL; |
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398 | module UNITR=freemodule(COL); |
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399 | matrix GCDR=UNITR; |
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400 | for(i=posdeg+1; i<=Sort; i++) |
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401 | { |
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402 | for(j=i+1; j<=Sort; j++) |
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403 | { |
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404 | GCDL=UNITL; |
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405 | GCDR=UNITR; |
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406 | G=gcd(STDM[i,i],STDM[j,j]); |
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407 | ideal Z=STDM[i,i],STDM[j,j]; |
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408 | matrix T=lift(Z,G); |
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409 | GCDL[i,i]=T[1,1]; |
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410 | GCDL[i,j]=T[2,1]; |
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411 | GCDL[j,i]=-STDM[j,j]/G; |
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412 | GCDL[j,j]=STDM[i,i]/G; |
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413 | GCDR[i,j]=T[2,1]*STDM[j,j]/G; |
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414 | GCDR[j,j]=T[2,1]*STDM[j,j]/G-1; |
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415 | GCDR[j,i]=1; |
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416 | STDM=GCDL*STDM*GCDR; |
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417 | TrafoLM=GCDL*TrafoLM; |
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418 | TrafoRM=TrafoRM*GCDR; |
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419 | } |
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420 | } |
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421 | list RUECK=TrafoRM; |
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422 | RUECK=insert(RUECK,STDM); |
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423 | RUECK=insert(RUECK,TrafoLM); |
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424 | return(RUECK); |
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425 | } |
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426 | |
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427 | |
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428 | // TODO: THIS PROCEDURE SEEMS TOO COMPLICATED, THUS PLEASE |
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429 | // TODO: 1. ADD HELP WITH ASSUMPTIONS AND RETURN DESCRIPTION |
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430 | // TODO: 2. ADD STEP-BY-STEP COMMENTS INTO THE CODE |
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431 | |
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432 | static proc diagonal_without_trafo( R, matrix MA, int B) |
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433 | { |
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434 | int ppl = printlevel-voice+2; |
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435 | |
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436 | int BASIS=B; |
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437 | int ROW=ncols(MA); |
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438 | int COL=nrows(MA); |
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439 | module m=MA[1]; |
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440 | int i; |
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441 | for(i=2;i<=ROW;i++) |
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442 | {m=m,MA[i];} |
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443 | |
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444 | |
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445 | list RINGLIST=ringlist(R); |
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446 | list o="C",0; |
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447 | list oo="lp",1; |
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448 | list ORD=o,oo; |
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449 | RINGLIST[3]=ORD; |
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450 | def r=ring(RINGLIST); |
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451 | setring r; |
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452 | //RICHTIGE ORDNUNG MACHEN |
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453 | map MAP=R,var(1); |
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454 | module m=MAP(m); |
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455 | |
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456 | int flag=1; ///////////////counts if the underlying ring is r (flag mod 2 == 1) or ro (flag mod 2 == 0) |
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457 | |
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458 | |
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459 | int act, j, ff; |
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460 | option(redSB); |
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461 | option(redTail); |
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462 | |
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463 | |
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464 | module STD=transpose(m); |
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465 | module TSTD; |
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466 | int finish=0; |
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467 | matrix STDFIN; |
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468 | STDFIN=STD; |
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469 | list COMPARE=STDFIN; |
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470 | |
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471 | while(finish==0) |
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472 | { |
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473 | dbprint(ppl,"Going into the while cycle"); |
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474 | dbprint(ppl,"Computing Groebner basis: start"); |
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475 | dbprint(ppl-1,STD); |
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476 | STD=engine(STD,BASIS); |
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477 | dbprint(ppl,"Computing Groebner basis: finished"); |
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478 | dbprint(ppl-1,STD); |
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479 | STDFIN=STD; |
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480 | /////////////////////////////////// check if the D-th row is finished /////////////////////////////////// |
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481 | COMPARE=insert(COMPARE,STDFIN); |
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482 | if(size(COMPARE)>=3) |
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483 | { |
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484 | if(STD==COMPARE[3]){finish=1;} |
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485 | } |
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486 | ////////////////////////////////// change to the opposite module |
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487 | |
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488 | TSTD=transpose(STD); |
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489 | STD=TSTD; |
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490 | flag++; |
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491 | dbprint(ppl,"Finished one while cycle"); |
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492 | } |
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493 | |
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494 | matrix STDMM=STD; |
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495 | list pos=list(); |
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496 | int fehlpos=0; |
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497 | while( size(STD)+fehlpos-ncols(STDMM) < 0) |
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498 | { |
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499 | for(i=1; i<=ncols(STDMM); i++) |
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500 | { |
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501 | ff=0; |
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502 | for(j=1; j<=nrows(STDMM); j++) |
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503 | { |
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504 | if (STD[j,i]==0) { ff++; } |
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505 | } |
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506 | if(ff==nrows(STDMM)) |
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507 | { |
---|
508 | pos=insert(pos,i); fehlpos++; |
---|
509 | } |
---|
510 | } |
---|
511 | } |
---|
512 | int fehlposc=fehlpos; |
---|
513 | module SORT; |
---|
514 | for(i=1; i<=fehlpos; i++) |
---|
515 | { |
---|
516 | SORT=gen(2); |
---|
517 | for (j=3;j<=ROW;j++) |
---|
518 | { |
---|
519 | SORT=SORT,gen(j); |
---|
520 | } |
---|
521 | SORT=SORT,gen(1); |
---|
522 | STD=STD*SORT; |
---|
523 | } |
---|
524 | |
---|
525 | //zero rows to the end |
---|
526 | STDMM=transpose(STD); |
---|
527 | pos=list(); |
---|
528 | fehlpos=0; |
---|
529 | while( size(transpose(STD))+fehlpos-ncols(STDMM) < 0) |
---|
530 | { |
---|
531 | for(i=1; i<=ncols(STDMM); i++) |
---|
532 | { |
---|
533 | ff=0; for(j=1; j<=nrows(STDMM); j++) |
---|
534 | { |
---|
535 | if(transpose(STD)[j,i]==0){ ff++;}} |
---|
536 | if(ff==nrows(STDMM)) { pos=insert(pos,i); fehlpos++; } |
---|
537 | } |
---|
538 | } |
---|
539 | int fehlposr=fehlpos; |
---|
540 | |
---|
541 | for(i=1; i<=fehlpos; i++) |
---|
542 | { |
---|
543 | SORT=gen(2); |
---|
544 | for(j=3;j<=COL;j++){SORT=SORT,gen(j);} |
---|
545 | SORT=SORT,gen(1); |
---|
546 | SORT=transpose(SORT); |
---|
547 | STD=SORT*STD; |
---|
548 | } |
---|
549 | |
---|
550 | //add zero rows or columns |
---|
551 | |
---|
552 | int adrow=COL-size(transpose(STD)); |
---|
553 | int adcol=ROW-size(STD); |
---|
554 | |
---|
555 | for(i=1;i<=adcol;i++){STD=STD,0;} |
---|
556 | STD=transpose(STD); |
---|
557 | for(i=1;i<=adrow;i++){STD=STD,0;} |
---|
558 | STD=transpose(STD); |
---|
559 | |
---|
560 | setring R; |
---|
561 | map MAPinv=r,var(1); |
---|
562 | module STD=MAPinv(STD); |
---|
563 | matrix STDM=STD; |
---|
564 | return(STDM); |
---|
565 | } |
---|
566 | |
---|
567 | |
---|
568 | |
---|
569 | static proc engine(module I, int i) |
---|
570 | { |
---|
571 | module J; |
---|
572 | if (i==0) |
---|
573 | { |
---|
574 | J = std(I); |
---|
575 | } |
---|
576 | if (i==1) |
---|
577 | { |
---|
578 | J = groebner(I); |
---|
579 | } |
---|
580 | if (i==2) |
---|
581 | { |
---|
582 | J = slimgb(I); |
---|
583 | } |
---|
584 | return(J); |
---|
585 | } |
---|
586 | |
---|
587 | /////////////////////////////////////////////////////////////////////////////// |
---|
588 | |
---|
589 | |
---|
590 | // TODO: IS THE ALGORITHM WELL KNOWN OR IS IT DUE TO SOME BOOK??? |
---|
591 | // ADD REFERENCE IF SO. |
---|
592 | // TODO: ADD A NOTE ABOUT THE CHOSEN JACOBSON NORMAL FORM! |
---|
593 | |
---|
594 | proc jacobson(matrix MA, list #) |
---|
595 | "USAGE: jacobson(M[, eng]); M matrix, eng an optional integer |
---|
596 | RETURN: list, a list containing matrices {U,D,V}, such that U*M*V = D, |
---|
597 | where U and V are unimodular, D is diagonal. |
---|
598 | ASSUME: The current ring is assumed to be the Ore localization of the first |
---|
599 | Weyl algebra with respect to S = K[x]\{0} (i.e. K(x)<D|D*x=x*D+1>). |
---|
600 | NOTE: The optional integer eng2 determines the engine, that computes the GB: |
---|
601 | @* 0 means 'std' (default), 1 means 'groebner' and 2 means 'slimgb'. |
---|
602 | @* If @code{printlevel}=1, progress debug messages will be printed, |
---|
603 | @* if @code{printlevel}>=2, all the debug messages will be printed. |
---|
604 | EXAMPLE: example jacobson; shows examples |
---|
605 | " |
---|
606 | { |
---|
607 | def R = basering; |
---|
608 | |
---|
609 | // TODO: ADD ASSUMPTION CHECKS!!! |
---|
610 | |
---|
611 | int B=0; |
---|
612 | |
---|
613 | if ( size(#)>0 ) |
---|
614 | { |
---|
615 | if (typeof(#[1])=="int") |
---|
616 | { |
---|
617 | B = #[1]; // zero can also happen |
---|
618 | } |
---|
619 | } |
---|
620 | |
---|
621 | //change ring |
---|
622 | list RINGLIST=ringlist(R); |
---|
623 | list o="C",0; |
---|
624 | intvec v=1,0; |
---|
625 | list oo="a",v; |
---|
626 | v=1,1; |
---|
627 | list ooo="lp",v; |
---|
628 | list ORD=o,oo,ooo; |
---|
629 | RINGLIST[3]=ORD; |
---|
630 | def r=ring(RINGLIST); |
---|
631 | setring r; |
---|
632 | |
---|
633 | //fix the required ordering |
---|
634 | matrix M=imap(R, MA); |
---|
635 | |
---|
636 | list T = triangle(M,B); |
---|
637 | module TrafoL = T[1]; |
---|
638 | module m = T[2]; |
---|
639 | module TrafoR = T[3]; |
---|
640 | |
---|
641 | //back to the ring |
---|
642 | setring R; |
---|
643 | |
---|
644 | matrix MAA = imap(r, m); |
---|
645 | matrix CON = divideByContent(MAA); |
---|
646 | matrix TL = imap(r, TrafoL); |
---|
647 | matrix TR = imap(r, TrafoR); |
---|
648 | |
---|
649 | // TODO: CHECK WHETHER HERE SHOULD BE CON*TR!? |
---|
650 | return(list(CON*TL, CON*MAA, TR)); |
---|
651 | |
---|
652 | } |
---|
653 | example |
---|
654 | { "EXAMPLE:"; echo = 2; |
---|
655 | ring r = 0,(x,d),Dp; |
---|
656 | def R=nc_algebra(1,1); setring R; |
---|
657 | R; // the Weyl algebra in x and d |
---|
658 | // jacobson(matrix(0)); // TODO: BUG HERE!!! |
---|
659 | |
---|
660 | matrix m[2][2]=d,x,0,d; |
---|
661 | print(m); |
---|
662 | list J = jacobson(m); // returns a list with 3 entries |
---|
663 | print(J[2]); // a Jacobson Form D |
---|
664 | // TODO: ARE U AND V REALLY UNIMODULAR? |
---|
665 | J[1]; // U |
---|
666 | J[3]; // V |
---|
667 | |
---|
668 | print(J[1]*m*J[3] - J[2]); // check that U*M*V = D |
---|
669 | |
---|
670 | matrix m[3][2]=x, x^4+x^2+21, x^4+x^2+x, x^3+x, 4*x^2+x, x; |
---|
671 | print(m); // M |
---|
672 | list JJ = jacobson(m); // returns a list with 3 entries |
---|
673 | |
---|
674 | // TODO: BUG: WHY THE FOLLOWING JACOBSON FORM IS NOT DIAGONAL???! |
---|
675 | print(JJ[2]); // a Jacobson Form D |
---|
676 | print(JJ[1]*m*JJ[3] - JJ[2]); // check that U*M*V = D |
---|
677 | |
---|
678 | // TODO: ARE U AND V REALLY UNIMODULAR? |
---|
679 | JJ[1]; // U |
---|
680 | JJ[3]; // V |
---|
681 | |
---|
682 | list S=smith(m,1); |
---|
683 | |
---|
684 | print(S[2]); // Smith Normal Form S of M |
---|
685 | print(S[1]*m*S[3] - S[2]); // check that U*M*V = S |
---|
686 | |
---|
687 | // TODO: ARE U AND V REALLY UNIMODULAR? |
---|
688 | S[1]; // U |
---|
689 | S[3]; // V |
---|
690 | } |
---|
691 | |
---|
692 | |
---|
693 | // TODO: THIS PROCEDURE SEEMS TOO COMPLICATED, THUS PLEASE |
---|
694 | // TODO: 1. ADD HELP WITH ASSUMPTIONS AND RETURN DESCRIPTION |
---|
695 | // TODO: 2. ADD STEP-BY-STEP COMMENTS INTO THE CODE |
---|
696 | |
---|
697 | static proc triangle( matrix MA, int B) |
---|
698 | { |
---|
699 | int ppl = printlevel-voice+2; |
---|
700 | |
---|
701 | map inv=ncdetection(); |
---|
702 | int ROW=ncols(MA); |
---|
703 | int COL=nrows(MA); |
---|
704 | |
---|
705 | //generate a module consisting of the columns of MA |
---|
706 | module m=MA[1]; |
---|
707 | int i,j,s,st,p,k,ff,ex, nz, ii,nextp; |
---|
708 | for(i=2;i<=ROW;i++) |
---|
709 | { |
---|
710 | m=m,MA[i]; |
---|
711 | } |
---|
712 | int BASIS=B; |
---|
713 | |
---|
714 | //add zero rows or columns |
---|
715 | int adrow=0; |
---|
716 | for(i=1;i<=COL;i++) |
---|
717 | { |
---|
718 | k=0; |
---|
719 | for(j=1;j<=ROW;j++) |
---|
720 | { |
---|
721 | if(MA[i,j]!=0){k=1;} |
---|
722 | } |
---|
723 | if(k==0){adrow++;} |
---|
724 | } |
---|
725 | |
---|
726 | m=transpose(m); |
---|
727 | for(i=1;i<=adrow;i++){m=m,0;} |
---|
728 | m=transpose(m); |
---|
729 | |
---|
730 | |
---|
731 | int flag=1; ///////////////counts if the underlying ring is r (flag mod 2 == 1) or ro (flag mod 2 == 0) |
---|
732 | |
---|
733 | module TrafoL=freemodule(COL); |
---|
734 | module TrafoR=freemodule(ROW); |
---|
735 | module EXL=freemodule(COL); //because we start with transpose(m) |
---|
736 | module EXR=freemodule(ROW); |
---|
737 | |
---|
738 | option(redSB); |
---|
739 | option(redTail); |
---|
740 | module STD_EX,LT,TSTD, L, Trafo; |
---|
741 | |
---|
742 | |
---|
743 | |
---|
744 | module STD=transpose(m); |
---|
745 | int finish=0; |
---|
746 | list pos, COM, COM_EX; |
---|
747 | matrix END, ENDSTD, STDFIN; |
---|
748 | STDFIN=STD; |
---|
749 | list COMPARE=STDFIN; |
---|
750 | |
---|
751 | |
---|
752 | while(finish==0) |
---|
753 | { |
---|
754 | dbprint(ppl,"Going into the while cycle"); |
---|
755 | if(flag mod 2==1){STD_EX=EXL,transpose(STD); ex=2*COL;} else {STD_EX=EXR,transpose(STD); ex=2*ROW;} |
---|
756 | |
---|
757 | dbprint(ppl,"Computing Groebner basis: start"); |
---|
758 | dbprint(ppl-1,STD); |
---|
759 | STD=engine(STD,BASIS); |
---|
760 | dbprint(ppl,"Computing Groebner basis: finished"); |
---|
761 | dbprint(ppl-1,STD); |
---|
762 | if(flag mod 2==1){s=ROW; st=COL;}else{s=COL; st=ROW;} |
---|
763 | |
---|
764 | STD_EX=transpose(STD_EX); |
---|
765 | dbprint(ppl,"Computing Groebner basis for transformation matrix: start"); |
---|
766 | dbprint(ppl-1,STD_EX); |
---|
767 | STD_EX=engine(STD_EX,BASIS); |
---|
768 | dbprint(ppl,"Computing Groebner basis for transformation matrix: finished"); |
---|
769 | dbprint(ppl-1,STD_EX); |
---|
770 | |
---|
771 | COM=1; |
---|
772 | COM_EX=1; |
---|
773 | for(i=2; i<=size(STD); i++) |
---|
774 | { COM=COM[1..size(COM)],i; COM_EX=COM_EX[1..size(COM_EX)],i; } |
---|
775 | nz=size(STD_EX)-size(STD); |
---|
776 | |
---|
777 | //zero rows in the begining |
---|
778 | |
---|
779 | if(size(STD)!=size(STD_EX) ) |
---|
780 | { |
---|
781 | for(i=1; i<=size(STD_EX)-size(STD); i++) |
---|
782 | { |
---|
783 | COM_EX=0,COM_EX[1..size(COM_EX)]; |
---|
784 | } |
---|
785 | } |
---|
786 | |
---|
787 | |
---|
788 | |
---|
789 | |
---|
790 | for(i=nz+1; i<=size(STD_EX); i++ ) |
---|
791 | {if( leadcoef(STD[i-nz])!=leadcoef(STD_EX[i]) ) {STD[i-nz]=leadcoef(STD_EX[i])*STD[i-nz];} |
---|
792 | } |
---|
793 | |
---|
794 | //assign the zero rows in STD_EX |
---|
795 | |
---|
796 | for (j=2; j<=nz; j++) |
---|
797 | { |
---|
798 | p=0; |
---|
799 | i=1; |
---|
800 | while(STD_EX[j-1][i]==0){i++;}; |
---|
801 | p=i-1; |
---|
802 | nextp=1; |
---|
803 | k=0; |
---|
804 | i=1; |
---|
805 | while(STD_EX[j][i]==0 and i<=p) |
---|
806 | { k++; i++;} |
---|
807 | if (k==p){ COM_EX[j]=-1; } |
---|
808 | } |
---|
809 | |
---|
810 | //assign the zero rows in STD |
---|
811 | for (j=2; j<=size(STD); j++) |
---|
812 | { |
---|
813 | i=size(transpose(STD)); |
---|
814 | while(STD[j-1][i]==0){i--;} |
---|
815 | p=i; |
---|
816 | i=size(transpose(STD[j])); |
---|
817 | while(STD[j][i]==0){i--;} |
---|
818 | if (i==p){ COM[j]=-1; } |
---|
819 | } |
---|
820 | |
---|
821 | for(j=1; j<=size(COM); j++) |
---|
822 | { |
---|
823 | if(COM[j]<0){COM=delete(COM,j);} |
---|
824 | } |
---|
825 | |
---|
826 | for(i=1; i<=size(COM_EX); i++) |
---|
827 | {ff=0; |
---|
828 | if(COM_EX[i]==0){ff=1;} |
---|
829 | else |
---|
830 | { for(j=1; j<=size(COM); j++) |
---|
831 | { if(COM_EX[i]==COM[j]){ff=1;} } |
---|
832 | } |
---|
833 | if(ff==0){COM_EX[i]=-1;} |
---|
834 | } |
---|
835 | |
---|
836 | L=STD_EX[1]; |
---|
837 | for(i=2; i<=size(COM_EX); i++) |
---|
838 | { |
---|
839 | if(COM_EX[i]!=-1){L=L,STD_EX[i];} |
---|
840 | } |
---|
841 | |
---|
842 | //////// split STD_EX in STD and the transformation matrix |
---|
843 | |
---|
844 | L=transpose(L); |
---|
845 | STD=L[st+1]; |
---|
846 | LT=L[1]; |
---|
847 | |
---|
848 | |
---|
849 | for(i=2; i<=s+st; i++) |
---|
850 | { |
---|
851 | if (i<=st) |
---|
852 | { |
---|
853 | LT=LT,L[i]; |
---|
854 | } |
---|
855 | if (i>st+1) |
---|
856 | { |
---|
857 | STD=STD,L[i]; |
---|
858 | } |
---|
859 | } |
---|
860 | |
---|
861 | STD=transpose(STD); |
---|
862 | STDFIN=matrix(STD); |
---|
863 | COMPARE=insert(COMPARE,STDFIN); |
---|
864 | LT=transpose(LT); |
---|
865 | |
---|
866 | ////////////////////// compute the transformation matrices |
---|
867 | |
---|
868 | if (flag mod 2 ==1) |
---|
869 | { |
---|
870 | TrafoL=transpose(LT)*TrafoL; |
---|
871 | } |
---|
872 | else |
---|
873 | { |
---|
874 | TrafoR=TrafoR*involution(LT,inv); |
---|
875 | } |
---|
876 | |
---|
877 | |
---|
878 | ///////////////////////// check whether the alg termined ///////////////// |
---|
879 | if(size(COMPARE)>=3) |
---|
880 | { |
---|
881 | if(STD==COMPARE[3]){finish=1;} |
---|
882 | } |
---|
883 | ////////////////////////////////// change to the opposite module |
---|
884 | TSTD=transpose(STD); |
---|
885 | STD=involution(TSTD,inv); |
---|
886 | flag++; |
---|
887 | dbprint(ppl,"Finished one while cycle"); |
---|
888 | } |
---|
889 | |
---|
890 | if (flag mod 2 ==0){ STD = involution(STD,inv);} else { STD = transpose(STD); } |
---|
891 | |
---|
892 | list REVERSE=TrafoL,STD,TrafoR; |
---|
893 | return(REVERSE); |
---|
894 | } |
---|
895 | |
---|
896 | static proc divideByContent(matrix M) |
---|
897 | { |
---|
898 | //find first entrie not equal to zero |
---|
899 | int i,k; |
---|
900 | k=1; |
---|
901 | vector CON; |
---|
902 | for(i=1;i<=ncols(M);i++) |
---|
903 | { |
---|
904 | if(leadcoef(M[i])!=0){CON=CON+leadcoef(M[i])*gen(k); k++;} |
---|
905 | } |
---|
906 | poly con=content(CON); |
---|
907 | matrix TL=1/con*freemodule(nrows(M)); |
---|
908 | return(TL); |
---|
909 | } |
---|
910 | |
---|
911 | |
---|
912 | /////interesting examples for smith//////////////// |
---|
913 | |
---|
914 | static proc Ex_One_wheeled_bicycle() |
---|
915 | { |
---|
916 | ring RA=(0,m), D, lp; |
---|
917 | matrix bicycle[2][3]=(1+m)*D^2, D^2, 1, D^2, D^2-1, 0; |
---|
918 | list s=smith(RA,bicycle, 1,0); |
---|
919 | print(s[2]); |
---|
920 | print(s[1]*bicycle*s[3]-s[2]); |
---|
921 | } |
---|
922 | |
---|
923 | |
---|
924 | static proc Ex_RLC-circuit() |
---|
925 | { |
---|
926 | ring RA=(0,m,R1,R2,L,C), D, lp; |
---|
927 | matrix circuit[2][3]=D+1/(R1*C), 0, -1/(R1*C), 0, D+R2/L, -1/L; |
---|
928 | list s=smith(RA,circuit, 1,0); |
---|
929 | print(s[2]); |
---|
930 | print(s[1]*circuit*s[3]-s[2]); |
---|
931 | } |
---|
932 | |
---|
933 | |
---|
934 | static proc Ex_two_pendula() |
---|
935 | { |
---|
936 | ring RA=(0,m,M,L1,L2,m1,m2,g), D, lp; |
---|
937 | |
---|
938 | matrix pendula[3][4]=m1*L1*D^2,m2*L2*D^2,(M+m1+m2)*D^2,-1,m1*L1^2*D^2-m1*L1*g,0,m1*L1*D^2,0,0, |
---|
939 | m2*L2^2*D^2-m2*L2*g,m2*L2*D^2,0; |
---|
940 | list s=smith(RA,pendula, 1,0); |
---|
941 | print(s[2]); |
---|
942 | print(s[1]*pendula*s[3]-s[2]); |
---|
943 | } |
---|
944 | |
---|
945 | |
---|
946 | |
---|
947 | static proc Ex_linerized_satellite_in_a_circular_equatorial_orbit() |
---|
948 | { |
---|
949 | ring RA=(0,m,omega,r,a,b), D, lp; |
---|
950 | |
---|
951 | matrix satellite[4][6]= |
---|
952 | D,-1,0,0,0,0, |
---|
953 | -3*omega^2,D,0,-2*omega*r,-a/m,0, |
---|
954 | 0,0,D,-1,0,0, |
---|
955 | 0,2*omega/r,0,D,0,-b/(m*r); |
---|
956 | |
---|
957 | list s=smith(RA,satellite, 1,0); |
---|
958 | print(s[2]); |
---|
959 | print(s[1]*satellite*s[3]-s[2]); |
---|
960 | } |
---|
961 | |
---|
962 | static proc Ex_flexible_one_link_robot() |
---|
963 | { |
---|
964 | ring RA=(0,M11,M12,M13,M21,M22,M31,M33,K1,K2), D, lp; |
---|
965 | |
---|
966 | matrix robot[3][4]=M11*D^2,M12*D^2,M13*D^2,-1,M21*D^2,M22*D^2+K1,0,0,M31*D^2,0,M33*D^2+K2,0; |
---|
967 | list s=smith(RA,robot, 1,0); |
---|
968 | print(s[2]); |
---|
969 | print(s[1]*robot*s[3]-s[2]); |
---|
970 | } |
---|
971 | |
---|
972 | |
---|
973 | |
---|
974 | /////interesting examples for jacobson//////////////// |
---|
975 | |
---|
976 | static proc Ex_compare_output_with_maple_package_JanetOre() |
---|
977 | { ring w = 0,(x,d),Dp; |
---|
978 | def W=nc_algebra(1,1); |
---|
979 | setring W; |
---|
980 | matrix m[3][3]=[d2,d+1,0],[d+1,0,d3-x2*d],[2d+1, d3+d2, d2]; |
---|
981 | list J=jacobson(W,m,0); |
---|
982 | print(J[1]*m*J[3]); |
---|
983 | print(J[2]); |
---|
984 | print(J[1]); |
---|
985 | print(J[3]); |
---|
986 | print(J[1]*m*J[3]-J[2]); |
---|
987 | } |
---|
988 | |
---|
989 | |
---|
990 | static proc Ex_cyclic() |
---|
991 | { ring w = 0,(x,d),Dp; |
---|
992 | def W=nc_algebra(1,1); |
---|
993 | setring W; |
---|
994 | matrix m[3][3]=d,0,0,x*d+1,d,0,0,x*d,d; |
---|
995 | list J=jacobson(W,m,0); |
---|
996 | print(J[1]*m*J[3]); |
---|
997 | print(J[2]); |
---|
998 | print(J[1]); |
---|
999 | print(J[3]); |
---|
1000 | print(J[1]*m*J[3]-J[2]); |
---|
1001 | } |
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1002 | |
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